generalized dominance relations.

positional_dominance(A, type = "one-mode", map = FALSE, benefit = TRUE)

A | Matrix containing attributes or relations, for instance calculated by indirect_relations. |
---|---|

type | A string which is either 'one-mode' (Default) if |

map | Logical scalar, whether rows can be sorted or not (Default). See Details. |

benefit | Logical scalar, whether the attributes or relations are benefit or cost variables. |

Dominance relations as matrix object. An entry `[u,v]`

is `1`

if u is dominated by v.

Positional dominance is a generalization of neighborhood-inclusion for
arbitrary network data. In the default case, it checks for all pairs \(u,v\) if
\(A_{ut} \ge A_{vt}\) holds for all \(t\) if `benefit = TRUE`

or
\(A_{ut} \le A_{vt}\) holds for all \(t\) if `benefit = FALSE`

.
This form of dominance is referred to as *dominance under total heterogeneity*.
If `map=TRUE`

, the rows of \(A\) are sorted decreasingly (`benefit = TRUE`

)
or increasingly (`benefit = FALSE`

) and then the dominance condition is checked. This second
form of dominance is referred to as *dominance under total homogeneity*, while the
first is called *dominance under total heterogeneity*.

Brandes, U., 2016. Network positions. *Methodological Innovations* 9,
2059799116630650.

Schoch, D. and Brandes, U., 2016. Re-conceptualizing centrality in social networks.
*European Journal of Applied Mathematics* 27(6), 971-985.

library(igraph) g <- graph.empty(n=11,directed = FALSE) g <- add_edges(g,c(1,11,2,4,3,5,3,11,4,8,5,9,5,11,6,7,6,8, 6,10,6,11,7,9,7,10,7,11,8,9,8,10,9,10)) P<-neighborhood_inclusion(g) comparable_pairs(P)#> [1] 0.1636364# positional dominance under total heterogeneity dist <- indirect_relations(g,type = "dist_sp") D <- positional_dominance(dist,map = FALSE,benefit = FALSE) comparable_pairs(D)#> [1] 0.1636364# positional dominance under total homogeneity D_map <- positional_dominance(dist,map = TRUE,benefit = FALSE) comparable_pairs(D_map) #more comparables than D#> [1] 0.8727273